Measurement-based preparation of stable coherent states of a Kerr parametric oscillator

Kerr parametric oscillators (KPOs) have attracted increasing attention in terms of their application to quantum information processing and quantum simulations. The state preparation and measurement of KPOs are typical requirements when used as qubits. The methods previously proposed for state preparations of KPOs utilize modulation of external fields such as a pump and drive fields. We study the stochastic state preparation of stable coherent states of a KPO with homodyne detection, which does not require modulation of external fields, and thus can reduce experimental efforts and exclude unwanted effects of possible imperfection in control of external fields. We quantitatively show that the detection data, if averaged over an optimal averaging time to decrease the effect of measurement noise, has a strong correlation with the state of the KPO, and therefore can be used to estimate the state (stochastic state preparation). We examine the success probability of the state estimation taking into account the measurement noise and bit flips. Moreover, the proper range of the averaging time to realize a high success probability is obtained by developing a binomial-coherent-state model, which describes the stochastic dynamics of the KPO under homodyne detection.


S1 Hamiltonian and master equation
The Hamiltonian for the composite system of a Kerr parametric oscillator (KPO) and a transmission line (TL) can be written asĤ tot =hω sâ †â −h χ 12 (â † +â) 4 + 2hβ (â † +â) 2 cos(ω p t) +h where ω s is the resonance frequency of the KPO when no pump filed is applied, andâ is the annihilation operator for the KPO. The second and third terms represent the anharmonicity of the KPO and the effect of the pump 1, 2 , respectively. β , ω p and χ are the amplitude and angular frequency of the pump and the anharmonicity parameter of the KPO, respectively. The fourth term is the Hamiltonian of the eigenmodes of the TL, and the fifth term is the interaction Hamiltonian between the KPO and TL. Here,b k is the annihilation operator of the mode with wave number k in the TL; v b is the phase velocity of the microwave in the TL; κ is the decay rate to the TL. We assume that the loss of microwave photons is negligible for simplicity.

S2 Average of jump interval
We obtain the average of the time interval between jumps, E[T i ], by using the binomial-coherent-state model. As explained in the main text, in this model, the KPO can only take either of |±α⟩ and jumps between the two states with a probability of p = Ωdt in time dt, where Ω is the rate of jumps. The average of the time interval between jumps is given by Eq. (9). Using

1/5
Eq. (11), the expected value ofx = (â +â † )/2 is represented as where we used N = t/dt. Taking the limit of dt → 0, we obtain (S10) We can obtain Ω by fitting ⟨x⟩ in Eq. (S10) to the counterpart of the dynamics governed by the master equation (S2). (Note that ρ(t) in Eq. (11) coincides with the solution of the master equation in Eq. (S2) when the binomial-coherent-state model is valid as explained in the main text.) Figure S1(a) presents the result of the fitting for α = 1.38 − 0.18i as an example. The time dependence of ⟨x⟩ in Eq. (S10) with Ω/2π = 20kHz matches well to the one obtained by solving the master equation (S2) for α = 1.38 − 0.18i. Figure S1(b) shows E[T i ] as a function of |α| 2 . It is seen that E[T i ] exponentially increases with the increase of |α| 2 3 . Figure S1(c) shows E[T i ] as a function of κ. It is seen that E[T i ] rapidly increases with the decrease of κ.

S3 Other protocols
We examine alternative methods to generate pure states of a KPO, a stable coherent state and a cat state.

Stable coherent state
Stable coherent states can be generated with a controlled single-photon drive field and a fixed pump field. The role of the drive field is to tilt the effective potential of a KPO 4 in order to increase the probability of realization of |α⟩ 5 . The time dependence of the Rabi frequency of the drive field is given by , dr . The Rabi frequency of the drive field is gradually ramped for 0 ≤ t ≤ T dr , and then is decreased to zero for T (2) dr . We set T (1) dr and T (2) dr to be long enough so that a stable coherent state is realized as the stationary state at t = T (2) dr . We define the success probability of the preparation by the fidelity, F [|α⟩⟨α| , ρ(T dr )], corresponding to the time when the drive field is off because normally a KPO is not subject to a drive field unless a single-qubit gate is operated. We simulate the dynamics of the KPO using the master equation represented as 1 where θ dr is the relative phase of the drive field to the one of the pump field 5 . We call θ dr the phase of the drive field for simplicity of notation. The initial state is the stationary state under the fixed pump field approximated by (|α⟩⟨α| + |−α⟩⟨−α|)/2. Typical time dependence of Ω and the corresponding fidelity are shown in Fig. S2(a). The fidelity tends to decrease for t > T (2) dr due to bit-flip, while there is a peak just after t = T (2) dr . We attribute this rise of the fidelity to that the coherent state pushed by the drive field becomes close to the target one temporally while Ω is decreased. The dependence of the success probability on ∆T dr is presented in Fig. S2(b,c) for various values of Ω 0 . It is seen that there is a proper range of ∆T dr to obtain high success probability. The maximum success probability is approximately 0.993 and is slightly higher than that of our simple measurement-based protocol presented in the main text. We attribute the decrease of the success probability in the small and large-∆T dr regimes to nonadiabatic transitions and bit-flip, respectively. Because the orientation of the tilting of the effective potential changes with θ dr , the success probability depends on θ dr . Figure S2(d) shows ∆T dr dependence of 1−success probability for various values of θ dr with Ω 0 /2π = 6 MHz. It is seen that the success probability can decrease when θ dr deviates from zero. Therefore, both ∆T dr and θ dr should be properly chosen for high success probability.
Although we study here the protocol which does not use control of the pump field, the stable coherent state can also be generated by controlling both the drive and pump fields 5 . .

Cat state
A cat state, which is a superposition of the stable coherent states, can be generated by gradually ramping the pump field against the vacuum state 1 . The cat state is represented as |Ψ cat ⟩ = N cat (|α⟩ + |−α⟩) with the normalization factor N cat . The ramp 4/5 of the pump field should be slow enough because the protocol has recourse to quantum adiabatic dynamics. We numerically examine the efficiency of the control. The time dependence of the pump amplitude β is given by where T p is the duration of the ramp of the pump field. The dynamics is simulated using the master equation (S2). The initial state is the vacuum state. A typical time dependence of the amplitude of the pump field and the corresponding fidelity F [|Ψ cat ⟩⟨Ψ cat | , ρ(t)] are exhibited in Fig. S3(a). The success probability is defined by the fidelity, F [|Ψ cat ⟩⟨Ψ cat | , ρ(T p )], corresponding to the time when the pump field is fixed. Figure S3(b) shows the dependence of the success probability on T p for various values of κ. The efficiency of the protocol is degraded by nonadiabatic transitions for small T p and by photon loss for large T p . The maximum success probability is lower than that of the measurement-based protocol for the stable coherent states. Although the success probability of creation of a cat state can be increased by tailoring the time dependence of detuning 6 , such control is out of the scope of this paper aiming to propose a simple method.